IODS course project
Minttu Lindholm
About the project
I went to the R-courses last fall and even though it was really difficult to me and lot of hard work needed, I really got excited and now Im here again feeling very uncertain but at the same time very motivated to learn. Here is the link to my IODS project in GitHub https://github.com/madmintt/IODS-project
REGRESSION AND MODEL VALIDATION
This week I have learned to draw advanced scatterplots and some new things about the diagnostics of regressionmodel.
##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2##
## Attaching package: 'GGally'## The following object is masked from 'package:dplyr':
##
## nasalearning2014 <-read.csv("create_learning2014.csv", TRUE, ",")
summary(learning2014)## X.3 X.2 X.1 X
## Min. : 1.00 Min. : 1.00 Min. : 1.00 Min. : 1.00
## 1st Qu.: 42.25 1st Qu.: 42.25 1st Qu.: 42.25 1st Qu.: 42.25
## Median : 83.50 Median : 83.50 Median : 83.50 Median : 83.50
## Mean : 83.50 Mean : 83.50 Mean : 83.50 Mean : 83.50
## 3rd Qu.:124.75 3rd Qu.:124.75 3rd Qu.:124.75 3rd Qu.:124.75
## Max. :166.00 Max. :166.00 Max. :166.00 Max. :166.00
## gender Age Attitude deep stra
## F:110 Min. :17.00 Min. :1.400 Min. :1.583 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:3.333 1st Qu.:2.625
## Median :22.00 Median :3.200 Median :3.667 Median :3.188
## Mean :25.51 Mean :3.143 Mean :3.680 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:4.083 3rd Qu.:3.625
## Max. :55.00 Max. :5.000 Max. :4.917 Max. :5.000
## surf Points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00This data includes 166 observations and seven variables, witch are gender, age, attitude (towards statistics) and three different studyingstrategies -or skills; deep, surface and strategic. The data includes also the exam points -variable, whitch we consider here the indicator of learning statistics.
VISUALIZING THE DATA
Plot matrix of the variables
visual_d <- ggpairs(learning2014, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
visual_d
pa <- ggplot(learning2014, aes(x = Attitude, y = Points, col = gender))
pat <- pa + geom_point()
patt <-pat + geom_smooth(method = "lm")
patti <-patt + ggtitle("Attitude vs. points")
patti
There seems to be a positive connection between attitude and points.
pd <- ggplot(learning2014, aes(x = deep, y = Points, col = gender))
pde <- pd + geom_point()
pdee <- pde + geom_smooth(method = "lm")
pdeep <- pdee + ggtitle("Deep vs. points")
pdeep
There seems to be absolutely no connection between deep and points.
pst <- ggplot(learning2014, aes(x = stra, y = Points, col = gender))
pstr <- pst + geom_point()
pstra <- pstr + geom_smooth(method = "lm")
pstrat <- pstra + ggtitle("Strategic vs. points")
pstrat
There might be a small positive connection between strategic and points.
psu <- ggplot(learning2014, aes(x = surf, y = Points, col = gender))
psur <- psu + geom_point()
psurf <- psur + geom_smooth(method = "lm")
psurfa <- psurf + ggtitle("Surface vs. points")
psurfa
There could be a small negative connection between surface and points.
ika <- ggplot(learning2014, aes(x = Age, y = Points, col = gender))
ikak <- ika + geom_point()
ikak <- ika + geom_smooth(method = "lm")
ikaka <- ikak + ggtitle("Age vs. points")
ikaka
According to this it seems like age has something to do with the points in the malegroup.
my_regressionmodel <- lm(formula = Points ~ Attitude, data = learning2014)
summary(my_regressionmodel)##
## Call:
## lm(formula = Points ~ Attitude, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.6372 1.8303 6.358 1.95e-09 ***
## Attitude 3.5255 0.5674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09I did the regression analysis first with having attitude, strategic and surface-learning as explanatory variables, because they seemed to be the three most potential explanatories when looking at the plots. However, the regression analysis showed, that no other variables but attitude are explaining the exampoints with statistical significance, so I only left that in the final model. So the interpretation is that when attitude increases one unit, the exampoints increases 3.5 in average. The multiple R-square is 0.1906 whitch means that the model explains 19 % of the variance of the exampoints in this case.
Diagnostic plots
Every statistical model includes several assumptions. The linear regresssion model assumes first of all that the errors of the model are normally distributed. Another assumption of linear regression model is that the errors have a constant variance and that they are not dependent of the explanatory variables. By making diagnostic plots we can see if theese assumptions come true in our model.
par(mfrow = c(2,2))
plot(my_regressionmodel, which = c(1, 2, 5))
By looking at the diagnostic plot of Residuals vs. Fitted values, we can see, that the plots form no pattern, whitch means that the errors do not depend on the explanatory variable,so no problem in there. The QQplot shows a nice fit with the line interpreting that the errors are normally distributed. The value of leverage is very low 0.04, which means that there are no observations with unusual leverage.
LOGISTICAL REGRESSION
library(ggplot2)
library(dplyr)
library(tidyr)
library(boot)alc <- read.csv("create_alc.csv", TRUE, ",")
summary(alc)## X school sex age address famsize
## Min. : 1.00 GP:342 F:198 Min. :15.00 R: 81 GT3:278
## 1st Qu.: 96.25 MS: 40 M:184 1st Qu.:16.00 U:301 LE3:104
## Median :191.50 Median :17.00
## Mean :191.50 Mean :16.59
## 3rd Qu.:286.75 3rd Qu.:17.00
## Max. :382.00 Max. :22.00
## Pstatus Medu Fedu Mjob Fjob
## A: 38 Min. :0.000 Min. :0.000 at_home : 53 at_home : 16
## T:344 1st Qu.:2.000 1st Qu.:2.000 health : 33 health : 17
## Median :3.000 Median :3.000 other :138 other :211
## Mean :2.806 Mean :2.565 services: 96 services:107
## 3rd Qu.:4.000 3rd Qu.:4.000 teacher : 62 teacher : 31
## Max. :4.000 Max. :4.000
## reason nursery internet guardian traveltime
## course :140 no : 72 no : 58 father: 91 Min. :1.000
## home :110 yes:310 yes:324 mother:275 1st Qu.:1.000
## other : 34 other : 16 Median :1.000
## reputation: 98 Mean :1.448
## 3rd Qu.:2.000
## Max. :4.000
## studytime failures schoolsup famsup paid activities
## Min. :1.000 Min. :0.0000 no :331 no :144 no :205 no :181
## 1st Qu.:1.000 1st Qu.:0.0000 yes: 51 yes:238 yes:177 yes:201
## Median :2.000 Median :0.0000
## Mean :2.037 Mean :0.2016
## 3rd Qu.:2.000 3rd Qu.:0.0000
## Max. :4.000 Max. :3.0000
## higher romantic famrel freetime goout
## no : 18 no :261 Min. :1.000 Min. :1.00 Min. :1.000
## yes:364 yes:121 1st Qu.:4.000 1st Qu.:3.00 1st Qu.:2.000
## Median :4.000 Median :3.00 Median :3.000
## Mean :3.937 Mean :3.22 Mean :3.113
## 3rd Qu.:5.000 3rd Qu.:4.00 3rd Qu.:4.000
## Max. :5.000 Max. :5.00 Max. :5.000
## Dalc Walc health absences
## Min. :1.000 Min. :1.000 Min. :1.000 Min. : 0.0
## 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:3.000 1st Qu.: 1.0
## Median :1.000 Median :2.000 Median :4.000 Median : 3.0
## Mean :1.482 Mean :2.296 Mean :3.573 Mean : 4.5
## 3rd Qu.:2.000 3rd Qu.:3.000 3rd Qu.:5.000 3rd Qu.: 6.0
## Max. :5.000 Max. :5.000 Max. :5.000 Max. :45.0
## G1 G2 G3 alc_use
## Min. : 2.00 Min. : 4.00 Min. : 0.00 Min. :1.000
## 1st Qu.:10.00 1st Qu.:10.00 1st Qu.:10.00 1st Qu.:1.000
## Median :12.00 Median :12.00 Median :12.00 Median :1.500
## Mean :11.49 Mean :11.47 Mean :11.46 Mean :1.889
## 3rd Qu.:14.00 3rd Qu.:14.00 3rd Qu.:14.00 3rd Qu.:2.500
## Max. :18.00 Max. :18.00 Max. :18.00 Max. :5.000
## high_use
## Mode :logical
## FALSE:268
## TRUE :114
##
##
## colnames(alc)## [1] "X" "school" "sex" "age" "address"
## [6] "famsize" "Pstatus" "Medu" "Fedu" "Mjob"
## [11] "Fjob" "reason" "nursery" "internet" "guardian"
## [16] "traveltime" "studytime" "failures" "schoolsup" "famsup"
## [21] "paid" "activities" "higher" "romantic" "famrel"
## [26] "freetime" "goout" "Dalc" "Walc" "health"
## [31] "absences" "G1" "G2" "G3" "alc_use"
## [36] "high_use"gather(alc) %>% glimpse## Warning: attributes are not identical across measure variables;
## they will be dropped## Observations: 13,752
## Variables: 2
## $ key <chr> "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "X", "...
## $ value <chr> "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11",...alc1 <- gather(alc) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free")## Warning: attributes are not identical across measure variables;
## they will be droppedalc1 + geom_bar()
This data approach student achievement in secondary education of two Portuguese schools and the variables include student grades, demographic, social and school related features. In my analysis here my task is to study witch variables affect students consumption of alcohol and predict it. My hypothesis is that studytime relates to alcohol consumption so that the more you drink the less time you study. Alcohol consumption could also have something to do with failures and absences. My guess is also that bad familyrelations could encrease alcohol consumption.
Barplot of alcohol consumption by sex
g1 <- ggplot(data = alc, aes(x = high_use, fill = sex))
g1 + geom_bar()
Ggplot of alcohol consumption
g2 <- ggplot(data = alc, aes(x = high_use))
g3 <- g2 + geom_bar()
g4 <- g3 + facet_wrap("sex")
g4
The distribution of familyrelations and alcohol consumption
g1 <- ggplot(alc, aes(x = high_use, y = famrel,col = sex))
g1 + geom_boxplot() + ylab("famrel")
alc %>% group_by(sex, high_use, famrel) %>% summarise(count = n())## # A tibble: 19 x 4
## # Groups: sex, high_use [4]
## sex high_use famrel count
## <fct> <lgl> <int> <int>
## 1 F FALSE 1 5
## 2 F FALSE 2 7
## 3 F FALSE 3 24
## 4 F FALSE 4 81
## 5 F FALSE 5 39
## 6 F TRUE 2 3
## 7 F TRUE 3 12
## 8 F TRUE 4 19
## 9 F TRUE 5 8
## 10 M FALSE 1 1
## 11 M FALSE 2 3
## 12 M FALSE 3 15
## 13 M FALSE 4 54
## 14 M FALSE 5 39
## 15 M TRUE 1 2
## 16 M TRUE 2 6
## 17 M TRUE 3 13
## 18 M TRUE 4 35
## 19 M TRUE 5 16By looking at the barplots it seems like there could be a relation between familyrelations and high use of alcohol; the familyrelations in the non-high use are higher. Same kind of evaluations seems possible when looking at the groopingtable as well.
The distributions of studytime and alcohol use
g1 <- ggplot(alc, aes(x = high_use, y = studytime,col = sex))
g1 + geom_boxplot() + ylab("studytime")
alc %>% group_by(sex, high_use, studytime) %>% summarise(count = n())## # A tibble: 16 x 4
## # Groups: sex, high_use [4]
## sex high_use studytime count
## <fct> <lgl> <int> <int>
## 1 F FALSE 1 18
## 2 F FALSE 2 83
## 3 F FALSE 3 39
## 4 F FALSE 4 16
## 5 F TRUE 1 9
## 6 F TRUE 2 25
## 7 F TRUE 3 7
## 8 F TRUE 4 1
## 9 M FALSE 1 40
## 10 M FALSE 2 52
## 11 M FALSE 3 13
## 12 M FALSE 4 7
## 13 M TRUE 1 33
## 14 M TRUE 2 35
## 15 M TRUE 3 1
## 16 M TRUE 4 3Studytime could also be connected to high use of alcohol, at least in femalegroups.
The distributions of failures and alcohol use
g1 <- ggplot(alc, aes(x = high_use, y = failures,col = sex))
g1 + geom_boxplot() + ylab("failures")
alc %>% group_by(sex, high_use, failures) %>% summarise(count = n())## # A tibble: 15 x 4
## # Groups: sex, high_use [4]
## sex high_use failures count
## <fct> <lgl> <int> <int>
## 1 F FALSE 0 143
## 2 F FALSE 1 8
## 3 F FALSE 2 5
## 4 F TRUE 0 33
## 5 F TRUE 1 7
## 6 F TRUE 2 1
## 7 F TRUE 3 1
## 8 M FALSE 0 101
## 9 M FALSE 1 4
## 10 M FALSE 2 5
## 11 M FALSE 3 2
## 12 M TRUE 0 57
## 13 M TRUE 1 5
## 14 M TRUE 2 8
## 15 M TRUE 3 2What it comes to failures, the hypothesis may not be true; the distributions are looking quite the same in both high-drinkers and those who are not drinking too much. When looking at the table, there are 144 heavy drinkers with zero failiors, but in the other groub the numerus is 90, witch is quite high as well.
The distributions of absences and alcohol use
g1 <- ggplot(alc, aes(x = high_use, y = absences,col = sex))
g1 + geom_boxplot() + ylab("absences")
alc %>% group_by(sex, high_use, absences) %>% summarise(count = n())## # A tibble: 66 x 4
## # Groups: sex, high_use [4]
## sex high_use absences count
## <fct> <lgl> <int> <int>
## 1 F FALSE 0 28
## 2 F FALSE 1 17
## 3 F FALSE 2 32
## 4 F FALSE 3 15
## 5 F FALSE 4 11
## 6 F FALSE 5 10
## 7 F FALSE 6 9
## 8 F FALSE 7 7
## 9 F FALSE 8 9
## 10 F FALSE 9 4
## # ... with 56 more rowsThe number of absences seems to be higher in the heavy drinking group.
A logistical regression model
Mymodel <- glm(high_use ~ failures + absences + famrel + studytime + sex, data = alc, family = "binomial")
summary(Mymodel)##
## Call:
## glm(formula = high_use ~ failures + absences + famrel + studytime +
## sex, family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.0557 -0.8196 -0.5880 1.0121 2.1090
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -0.08166 0.64708 -0.126 0.899579
## failures 0.33723 0.19336 1.744 0.081155 .
## absences 0.08492 0.02253 3.769 0.000164 ***
## famrel -0.27215 0.12898 -2.110 0.034855 *
## studytime -0.33349 0.16253 -2.052 0.040186 *
## sexM 0.85149 0.25493 3.340 0.000837 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 415.39 on 376 degrees of freedom
## AIC: 427.39
##
## Number of Fisher Scoring iterations: 4coef(Mymodel)## (Intercept) failures absences famrel studytime sexM
## -0.08165709 0.33722680 0.08491968 -0.27215425 -0.33349374 0.85149451OR <- coef(Mymodel) %>% exp
CI <- confint(Mymodel) %>% exp## Waiting for profiling to be done...cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.9215879 0.2570726 3.2760808
## failures 1.4010568 0.9596267 2.0600366
## absences 1.0886296 1.0436932 1.1405269
## famrel 0.7617368 0.5907516 0.9813904
## studytime 0.7164164 0.5166676 0.9790799
## sexM 2.3431461 1.4282690 3.8884700Acording to this model all explanatory variables with the exception of failures, explain alcohol consumption of the students with statistical significance. Even so, failures OD is 1.40 and the interval of confidence is between 0.96 and 2.06 so I dont know how to interpret this when OD over 1 means positive association with high drinking and it would be 1.4 times likely to be a heavy drinker when having one unit more absences.So now I really dont know if I should leave failures from my prediction. Another predictor seems to be absences with oddsratio 1.09, witch means that those with more absences are 1.9 times more likely to be high users. The confidence interval is between 1.04 and 1.14 witch is the narrowest in this model. The OR is over 1 in any case, witch means that absences are positively associated with heavy drinking. Sex seems to be a good explainer so that males are 2.34 times more likely (than females) to be high users of alcohol. Familyrelations and sudytime has weaker OR and the maximum in the interval of confidence is below 1 witch is logical since the better familyrelations you have and the more you study the lower odds for you to be a high user of alcohol.
Prediction
Mymodel <- glm(high_use ~ absences + sex + famrel + studytime, data = alc, family = "binomial")
probabilities <- predict(Mymodel, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probabilities > 0.5)
table(high_use = alc$high_use, prediction = probabilities > 0.5)## prediction
## high_use FALSE TRUE
## FALSE 250 18
## TRUE 86 28g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
table(high_use = alc$high_use, prediction = alc$prediction) %>% prop.table() %>% addmargins()## prediction
## high_use FALSE TRUE Sum
## FALSE 0.65445026 0.04712042 0.70157068
## TRUE 0.22513089 0.07329843 0.29842932
## Sum 0.87958115 0.12041885 1.00000000loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2722513The propability that the model predicts right FALSE is 0.65 and wronng FALSE 0.05. The propability that the model predicts right TRUE is 0.07 and wrong TRUE 0.23. The training error is 0.27.
cv <- cv.glm(data = alc, cost = loss_func, glmfit = Mymodel, K = 10)
cv$delta## [1] 0.2774869 0.2814547The training error is just a little bit bigger than in the model introduced in Datacamp.
Super-bonus
vari27 <- 0.194
vari26 <- 0.199
vari25 <- 0.196
vari24 <- 0.188
vari23 <- 0.191
vari22 <- 0.188
vari21 <- 0.199
vari20 <- 0.196
vari19 <- 0.196
vari18 <- 0.191
vari17 <- 0.196
vari16 <- 0.191
vari15 <- 0.194
vari14 <- 0.188
vari13 <- 0.188
vari12 <- 0.191
vari11 <- 0.196
vari10 <- 0.215
vari9 <- 0.220
vari8 <- 0.223
vari7 <- 0.217
vari6 <- 0.225
vari5 <- 0.262
vari4 <- 0.272
vari3 <- 0.257
vari2 <- 0.283
vari1 <- 0.288I just put all the variables in the model and started to remove the variables one of a time and then checked out the training error. THe trend here is that less variables, the bigger penalty.
variab <- c(vari1, vari2, vari3, vari4, vari5, vari6, vari7, vari8, vari9, vari10, vari11, vari12, vari13, vari14, vari15, vari16, vari17, vari18, vari19, vari20, vari21, vari22, vari23, vari24, vari25, vari26, vari27)
variables <- c( 1 : 27)
plot(variab, variables)
CLUSTERING AND CLASSIFICATION
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata("Boston")
library("ggplot2")
library("dplyr")
library("GGally")
library("corrplot")## corrplot 0.84 loadedstr(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00The Boston data includes variables describing Boston such as for example what is the degree of criminality, the distribution of age in Boston, lower status of population or pupil-teacher ratio in town. The data has 506 observations and 14 variables.
Bosgraph <- ggpairs(Boston, mapping = aes(alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
Bosgraph
cor(Boston)## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593
## ptratio black lstat medv
## crim 0.2899456 -0.38506394 0.4556215 -0.3883046
## zn -0.3916785 0.17552032 -0.4129946 0.3604453
## indus 0.3832476 -0.35697654 0.6037997 -0.4837252
## chas -0.1215152 0.04878848 -0.0539293 0.1752602
## nox 0.1889327 -0.38005064 0.5908789 -0.4273208
## rm -0.3555015 0.12806864 -0.6138083 0.6953599
## age 0.2615150 -0.27353398 0.6023385 -0.3769546
## dis -0.2324705 0.29151167 -0.4969958 0.2499287
## rad 0.4647412 -0.44441282 0.4886763 -0.3816262
## tax 0.4608530 -0.44180801 0.5439934 -0.4685359
## ptratio 1.0000000 -0.17738330 0.3740443 -0.5077867
## black -0.1773833 1.00000000 -0.3660869 0.3334608
## lstat 0.3740443 -0.36608690 1.0000000 -0.7376627
## medv -0.5077867 0.33346082 -0.7376627 1.0000000cor_matrix<-cor(Boston)
cor_matrix <- cor_matrix %>% round(digits = 2)
corrplot(cor_matrix, method= "circle")
In this correlation matrix we can see all the variables in the Boston data and their between-correlations. The more darker the color, the stronger the correlation. The shades of red describe negative, and the shades of blue positive correlation. The median value of owner-occupied homes has a strong negative correlation with lower status of population.
Standardizing the data and changing the continuous crime rate -variable in to a categorical one
Next the data will be standardized, ie. it will be scaled so that the column means are subtracted from the corresponding columns and then divided the difference with standard deviation.
boston_scaled <- scale(Boston)
summary(boston_scaled)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865class(boston_scaled)## [1] "matrix"boston_scaled <- as.data.frame(boston_scaled)summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label= c("low", "med_low", "med_high", "high"))
table(crime)## crime
## low med_low med_high high
## 127 126 126 127boston_scaled <- dplyr::select(boston_scaled, -crim)
boston_scaled <- data.frame(boston_scaled, crime)Splitting the data to train and test sets
n <- nrow(boston_scaled)
ind <- sample(n, size = n * 0.8)
train <- boston_scaled[ind,]
test <- boston_scaled[-ind,]
correct_classes <- test$crime
test <- dplyr::select(test, -crime)The LDA model
lda.fit <- lda(crime ~ ., data = train)
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2524752 0.2574257 0.2450495 0.2450495
##
## Group means:
## zn indus chas nox rm
## low 1.06369310 -0.9783716 -0.07933396 -0.9135729 0.42033731
## med_low -0.06877583 -0.2852721 -0.08304540 -0.5704280 -0.13928661
## med_high -0.38242941 0.1636528 0.20489520 0.4329879 0.06864948
## high -0.48724019 1.0149946 -0.07348562 1.0479694 -0.37214118
## age dis rad tax ptratio
## low -0.9394522 0.9321151 -0.6744874 -0.7249343 -0.43729048
## med_low -0.3625265 0.3650863 -0.5489875 -0.4749390 -0.08605428
## med_high 0.4757839 -0.3935452 -0.4308390 -0.3388924 -0.39330929
## high 0.8339088 -0.8479259 1.6596029 1.5294129 0.80577843
## black lstat medv
## low 0.38387345 -0.79191617 0.50820958
## med_low 0.34421756 -0.12798218 -0.02165382
## med_high 0.02596325 0.06141285 0.16864353
## high -0.85153308 0.89730985 -0.76456634
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.119794642 0.633434396 -0.913707123
## indus 0.055135832 -0.292486511 0.515680032
## chas -0.017231502 0.010844617 0.005481306
## nox 0.229695799 -0.750742048 -1.301856239
## rm 0.021115768 0.008407438 -0.094747201
## age 0.279772086 -0.436883239 -0.261737559
## dis -0.152929939 -0.301537656 0.163741351
## rad 3.670164754 0.849467899 -0.050775720
## tax -0.010186067 0.137922525 0.448772408
## ptratio 0.150775335 0.009661024 -0.281587785
## black -0.149088142 0.055343724 0.216445329
## lstat 0.090199748 -0.224988187 0.376666163
## medv 0.004867651 -0.508151854 -0.223352108
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9509 0.0376 0.0115lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(train$crime)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
lda.pred <- predict(lda.fit, newdata = test)
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 9 14 2 0
## med_low 2 16 4 0
## med_high 0 9 16 2
## high 0 0 1 27Here we can see that predicting is quite good since the rates are highest in the groups witch are correctly predicted and no radical errors exist.
dist_eu <- dist(boston_scaled)## Warning in dist(boston_scaled): NAs introduced by coercionsummary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1394 3.5267 4.9081 4.9394 6.2421 13.0045dist_man <- dist(boston_scaled, method = "manhattan")## Warning in dist(boston_scaled, method = "manhattan"): NAs introduced by
## coercionsummary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2849 8.8386 13.0300 13.8657 18.0953 46.8948km <-kmeans(Boston, centers = 2)
km$cluster## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2
## 505 506
## 2 2pairs(boston_scaled[1:5], col = km$cluster)
pairs(boston_scaled[6:10], col = km$cluster)
pairs(boston_scaled[11:14], col = km$cluster)
set.seed(123)
k_max <- (k_max = 10)
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
qplot(x = 1:k_max, y = twcss, geom = 'line')
I tried the model with 5,4,3 and 2 clusters and my interpretation when viewing the plots is that 3 is best but when calculating WCSS it seems that 2 clusters seems to be better choice. Anyhow variables tax (full-value property-tax rate per $10,000), rad (index of accessibility to radial highways) and some of the classes of crime seems to set clearly in clusters in any case.
Bonus
Bclust <-kmeans(Boston, centers = 3)
Bclust$cluster## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 3 3 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## 3 3 3 3 3 3 3 3 3 3 1 1 1 1 1 1 1 1
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 3 3 3 3 3 3 3 1 1 1 1 1 1 3 3 3 3 3
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 3 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306
## 3 3 3 3 3 3 3 3 3 3 1 1 1 3 3 3 3 3
## 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
## 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342
## 3 3 3 3 1 1 1 3 3 3 3 3 3 3 3 3 3 3
## 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360
## 1 1 1 1 1 3 3 3 3 1 1 3 3 3 2 2 2 2
## 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486
## 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
## 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504
## 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 3 3 3
## 505 506
## 3 3pairs(boston_scaled[1:5], col = Bclust$cluster)
pairs(boston_scaled[6:10], col = Bclust$cluster)
pairs(boston_scaled[11:14], col = Bclust$cluster)
lda.fit <- lda(Bclust$cluster ~ ., data = boston_scaled)
lda.fit## Call:
## lda(Bclust$cluster ~ ., data = boston_scaled)
##
## Prior probabilities of groups:
## 1 2 3
## 0.1996047 0.2707510 0.5296443
##
## Group means:
## zn indus chas nox rm age
## 1 -0.07332724 0.2818828 0.0005392655 0.2816899 -0.1453417 0.1822823
## 2 -0.48724019 1.0662784 -0.0424254043 0.9959393 -0.3962652 0.7599946
## 3 0.27670879 -0.6513071 0.0214843827 -0.6152775 0.2573427 -0.4572006
## dis rad tax ptratio black lstat
## 1 -0.2378455 -0.5418150 -0.01444889 -0.3768823 0.07010933 0.01371321
## 2 -0.8265965 1.5757732 1.53915759 0.8040926 -0.71893398 0.84321670
## 3 0.5121870 -0.6013344 -0.78136288 -0.2690134 0.34109296 -0.43621538
## medv crimemed_low crimemed_high crimehigh
## 1 -0.03812375 0.32673267 0.46534653 0.00990099
## 2 -0.68070813 0.03649635 0.04379562 0.91970803
## 3 0.36234147 0.32835821 0.27238806 0.00000000
##
## Coefficients of linear discriminants:
## LD1 LD2
## zn 0.05280738 -0.014188634
## indus 0.52954210 0.136232438
## chas -0.03744336 0.041429998
## nox 0.20190265 -0.583371775
## rm 0.02768863 0.117101924
## age -0.01312130 0.006188618
## dis -0.36742893 0.203235111
## rad 2.17313453 2.225891425
## tax 5.27359727 -2.772125460
## ptratio 0.14183368 0.150571551
## black -0.04900742 0.092304730
## lstat 0.19460083 0.100047863
## medv 0.32392299 -0.139204299
## crimemed_low -0.95555999 -0.081053964
## crimemed_high -2.22131989 0.172165534
## crimehigh -0.91590244 2.292286225
##
## Proportion of trace:
## LD1 LD2
## 0.9855 0.0145lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
classes <- as.numeric(Bclust$cluster)
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
Here we can see that the most influencial linear separatorvariables for the clusters are crimerates, rad and tax. LD1 explains 99% of the between group variance.
Superbonus
lda.fit <- lda(crime ~ ., data = train)
model_predictors <- dplyr::select(train, -crime)
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)library("plotly")##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)Bclust <-kmeans(Boston, centers = 3)
lda.fit <- lda(Bclust$cluster ~ ., data = boston_scaled)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = Bclust)DIMENSION REDUCTION TECHNIQUES
library("stringr")
library(GGally)
library(dplyr)
library(corrplot)
library(ggplot2)
library(tidyr)
library("FactoMineR")human_ <- read.csv("create_human_.csv", TRUE, ",")
str(human_)## 'data.frame': 155 obs. of 9 variables:
## $ X : Factor w/ 155 levels "Afghanistan",..: 105 6 134 41 101 54 67 149 28 102 ...
## $ labRatio : num 0.891 0.819 0.825 0.884 0.829 ...
## $ edu2Ratio: num 0.993 1.003 1.017 1.012 1.032 ...
## $ LifeExp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ ExpEdu : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ GNI : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ Matmort : int 4 6 6 5 6 7 9 28 11 8 ...
## $ AdolBR : num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ ParlPres : num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...dim(human_)## [1] 155 9summary(human_)## X labRatio edu2Ratio LifeExp
## Afghanistan: 1 Min. :0.1857 Min. :0.6681 Min. :49.00
## Albania : 1 1st Qu.:0.5984 1st Qu.:1.0032 1st Qu.:66.30
## Algeria : 1 Median :0.7535 Median :1.0667 Median :74.20
## Argentina : 1 Mean :0.7074 Mean :1.3523 Mean :71.65
## Armenia : 1 3rd Qu.:0.8535 3rd Qu.:1.3766 3rd Qu.:77.25
## Australia : 1 Max. :1.0380 Max. :5.8235 Max. :83.50
## (Other) :149
## ExpEdu GNI Matmort AdolBR
## Min. : 5.40 Min. : 581 Min. : 1.0 Min. : 0.60
## 1st Qu.:11.25 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65
## Median :13.50 Median : 12040 Median : 49.0 Median : 33.60
## Mean :13.18 Mean : 17628 Mean : 149.1 Mean : 47.16
## 3rd Qu.:15.20 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95
## Max. :20.20 Max. :123124 Max. :1100.0 Max. :204.80
##
## ParlPres
## Min. : 0.00
## 1st Qu.:12.40
## Median :19.30
## Mean :20.91
## 3rd Qu.:27.95
## Max. :57.50
## rownames(human_) <- human_$X
human_ <- select(human_, -X)Visualizing the data
ggpairs(human_)
cor(human_)%>%corrplot
This data is a joined data set, witch includes variables from HDI (Human development index) and GII (Gender inequality index). The HDI measures human development by three dimensions; Health, learning/education and standard of living. The variables of GII data measures the inequality between genders giving numbers for example the differences in empowerment, health and labour force. This combined dataset has 195 observations and 19 variables. When looking at the distributions of the variables in this data, we can see that non of the variables really seems to be normally distributed. By just looking at the picture maybe the womens participation in parliament and expected years of education could be normal, other distributions seems to be clearly skew. What comes to correlations between the variables, almost all the variables seems to have a strong connection between, except for womens participation in parliament witch seems to have smaller correlation with other variables in the data. We`ll see how the correlations will change when the amount of women will increace in parliaments hopefully in the future ;) No correlation seems to be between the ratio of labour force and the ratio of secondary education between males and females.
Principal components analysis with scaled and notscaled data
pca_human <- prcomp(human_)
biplot(pca_human, choices = 1:2,cex = c(0.5, 1), col = c("grey40", "deeppink2"))## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
## Warning in arrows(0, 0, y[, 1L] * 0.8, y[, 2L] * 0.8, col = col[2L], length
## = arrow.len): zero-length arrow is of indeterminate angle and so skipped
human_std <- scale(human_)
pca_human <- prcomp(human_std)
biplot(pca_human, choices = 1:2, cex = c(0.5, 1), col = c("grey40", "deeppink2"))
The data is the same in both cases of course, but with unscaled data PCA is sensitive to the relative scaling of the variance of the original variables,so the biplots of PCA looks quite different with scaled and unscaled data. With unscaaled data the PCA has all the other variables in one arrow pointing to PC1 dimension and demonstrating a perfect correlation between them as well as with PC1. This is not surprizing since the correlation between the variables is overall strong and with unscaled data the PCA has difficulties to differ the dimensions. Only the variable GNI has no arrow. Whit the scaled data we can see that the mortality rate of birthgiven mothers and adolescents birth rate has a strong connection with each other, witch was seen in the correlation matrix as well. Mothers mortality and adolescents birth rate seems to set nicely to dimension PC1. Expected years of education, national income per capita, life expectancy and the secondary education ratio between females and males seems to set also to dimension PC1 but not having correlation with mothers mortality rate and adolescens birth rate. The PC1 dimension seems to represent the health and knowledge -part of human development. On the second dimension in PCA there is women participation in parliament and the labor force ratio between females and males, witch has a between correlation as well. This we can consider as the empowerment -part of human development.
MCA
data("tea")keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
tea_time <- select(tea,one_of(keep_columns) )
dim(tea_time)## [1] 300 6str(tea_time)## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))## Warning: attributes are not identical across measure variables;
## they will be dropped
The tea data is about the tea consumption and the relating culture or habits. Here we look at 6 factor variables and the data consist of 300 observations.
mca <- MCA(tea_time, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |plot(mca, invisible=c("ind"), habillage = "quali")
mca <- MCA(tea_time, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |plot(mca, invisible=c("var"), habillage = "quali")
mca <- MCA(tea_time, graph = FALSE)
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6
## Variance 0.279 0.261 0.219 0.189 0.177 0.156
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953
## Dim.7 Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.144 0.141 0.117 0.087 0.062
## % of var. 7.841 7.705 6.392 4.724 3.385
## Cumulative % of var. 77.794 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898
## cos2 v.test Dim.3 ctr cos2 v.test
## black 0.003 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 0.027 2.867 | 0.433 9.160 0.338 10.053 |
## green 0.107 -5.669 | -0.108 0.098 0.001 -0.659 |
## alone 0.127 -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 0.035 3.226 | 1.329 14.771 0.218 8.081 |
## milk 0.020 2.422 | 0.013 0.003 0.000 0.116 |
## other 0.102 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag 0.161 -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 0.478 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged 0.141 -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |plot(mca, habillage = "quali", selectMod = "cos2 20", cex = 0.8)
Here we can see three biplots of MCA, where in one is observations or individuals modelled in dimensions and in the other variables modelled in dimensions. In third picture there are both. Here we can see for example that the unpacked, green tea and tea shop are clearly in the same dimension and milk, Earl gray, tea bag and chain store are in the second. To interpret this more deeply one should understand more the phenomenon and tea drinking human :)
ANALYZING THE LONGITUDINAL DATA
Graphical displays and summary measure approach with the RATS data
library(ggplot2)
library(dplyr)
library(tidyr)
library(lme4)## Loading required package: Matrix##
## Attaching package: 'Matrix'## The following objects are masked from 'package:tidyr':
##
## expand, pack, unpackRATSL <- read.csv("create_RATSL.csv", TRUE, ",")
glimpse(RATSL)## Observations: 176
## Variables: 6
## $ X <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ ID <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
## $ Group <int> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1...
## $ WD <fct> WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, WD1, ...
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, ...
## $ Time <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8...summary(RATSL)## X ID Group WD
## Min. : 1.00 Min. : 1.00 Min. :1.00 WD1 :16
## 1st Qu.: 44.75 1st Qu.: 4.75 1st Qu.:1.00 WD15 :16
## Median : 88.50 Median : 8.50 Median :1.50 WD22 :16
## Mean : 88.50 Mean : 8.50 Mean :1.75 WD29 :16
## 3rd Qu.:132.25 3rd Qu.:12.25 3rd Qu.:2.25 WD36 :16
## Max. :176.00 Max. :16.00 Max. :3.00 WD43 :16
## (Other):80
## Weight Time
## Min. :225.0 Min. : 1.00
## 1st Qu.:267.0 1st Qu.:15.00
## Median :344.5 Median :36.00
## Mean :384.5 Mean :33.55
## 3rd Qu.:511.2 3rd Qu.:50.00
## Max. :628.0 Max. :64.00
## First plotting the weights of the 16 rats with three different nutrition during 11 weeks.
RATSL$ID <- factor(RATSL$ID)
RATSL$Group <- factor(RATSL$Group)ggplot(RATSL, aes(x = Time, y = Weight, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(RATSL$Weight), max(RATSL$Weight)))
In the picture we can see first of all that the weight of every rat has grown through the weeks and in the groups two and three it seems like the bigger the rat has been in the beginning, the bigger it is also in the end. There is clearly a difference between the group 1 and the other groups but we cant be sure why is that because rats in the first group have been smaller in the beginning. In group one the growth seems to be quite steady. Also the variability seems to stay quite steady during the time period in every group but some differences occur among the rats but there cant really be seen clear differences between the groups. Lets see if standardizing the data helps us with that.
Standardized data
RATSL <- RATSL %>%
group_by(Time) %>%
mutate(stdratsl = (Weight - mean(Weight))/sd(Weight) ) %>%
ungroup()ggplot(RATSL, aes(x = Time, y = stdratsl, linetype = ID)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ Group, labeller = label_both) +
scale_y_continuous(name = "standardized bprs")
Here we can see, that some of the weight curves seems to be downward. Over all no clear trends can be seen.
Summary measure analysis
n <- RATSL$Time %>% unique() %>% length()
RATSS <- RATSL %>%
group_by(Group, Time) %>%
summarise( mean = mean(Weight), se = sd(Weight)/ sqrt(n) ) %>%
ungroup()
glimpse(RATSS)## Observations: 33
## Variables: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,...
## $ Time <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, ...
## $ mean <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 26...
## $ se <dbl> 4.589478, 3.947710, 3.460116, 4.100800, 3.333956, 3.5529...ggplot(RATSS, aes(x = Time, y = mean, linetype = Group, shape = Group)) +
geom_line() +
scale_linetype_manual(values = c(1,2,3)) +
geom_point(size=3) +
scale_shape_manual(values = c(1,2,3)) +
geom_errorbar(aes(ymin=mean-se, ymax=mean+se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.8,0.8)) +
scale_y_continuous(name = "mean(Weight) +/- se(Weight)")## Warning in if (position != "none") {: the condition has length > 1 and only
## the first element will be used## Warning in if (position == "manual") {: the condition has length > 1 and
## only the first element will be used## Warning in if (position == "left") {: the condition has length > 1 and only
## the first element will be used## Warning in if (position == "right") {: the condition has length > 1 and
## only the first element will be used## Warning in if (position == "bottom") {: the condition has length > 1 and
## only the first element will be used## Warning in if (position == "top") {: the condition has length > 1 and only
## the first element will be used## Warning in if (position == "manual") {: the condition has length > 1 and
## only the first element will be used
The summary measure method captures the variability of each observation, ie. the variability of the weight of each rat during the timepoints and now we can see, that there is not a difference between the groups two and three. Group one seems to differ from the other groups.
Boxplots and outliers
RATSL8S <- RATSL %>%
filter(Time > 0) %>%
group_by(Group, ID) %>%
summarise( mean=mean(Weight) ) %>%
ungroup()
glimpse(RATSL8S)## Observations: 16
## Variables: 3
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ mean <dbl> 261.0909, 237.6364, 260.1818, 266.5455, 269.4545, 274.72...ggplot(RATSL8S, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(Time), WD 1-11")
RATSL8S1 <- RATSL8S %>%
filter(mean < 550)ggplot(RATSL8S1, aes(x = Group, y = mean)) +
geom_boxplot() +
stat_summary(fun.y = "mean", geom = "point", shape=23, size=4, fill = "white") +
scale_y_continuous(name = "mean(Time), WD 1-11")
Here we have two boxplots; the first one showing the outlier in the data in group two, witch was actually quite clearly seen in the previous pictures as well. When removing the outlier, we can see that the variability seems to be smaller in group three than group two witch still doesnt have to mean that there is differences between the nutritions. In group one theres nothing new revealed here. To get closer to the truth about the diffeences between the groups, we will see how regression model works.
RATS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", header = TRUE, sep = '\t')
RATSL8S2 <- RATSL8S %>%
mutate(baseline = RATS$WD1)
fit <- lm(formula (mean ~ RATS$WD1 + Group), data = RATSL8S2)
summary(fit)##
## Call:
## lm(formula = formula(mean ~ RATS$WD1 + Group), data = RATSL8S2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -21.732 -3.812 1.991 6.889 13.455
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.14886 19.88779 1.516 0.1554
## RATS$WD1 0.93194 0.07793 11.959 5.02e-08 ***
## Group2 31.68866 17.11189 1.852 0.0888 .
## Group3 21.52296 21.13931 1.018 0.3287
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.62 on 12 degrees of freedom
## Multiple R-squared: 0.9947, Adjusted R-squared: 0.9933
## F-statistic: 747.8 on 3 and 12 DF, p-value: 6.636e-14Here we can see that there are no significant differences between the nutrition groups, but the weight in the beginnig is a statistically significant predictor.
Linear mixed effects models for normal response variables with the BPRSL data
BPRSL <-read.csv("create_BPRSL.csv", TRUE, ",")
dim(BPRSL)## [1] 360 6str(BPRSL)## 'data.frame': 360 obs. of 6 variables:
## $ X : int 1 2 3 4 5 6 7 8 9 10 ...
## $ treatment: int 1 1 1 1 1 1 1 1 1 1 ...
## $ subject : int 1 2 3 4 5 6 7 8 9 10 ...
## $ weeks : Factor w/ 9 levels "week0","week1",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ bprs : int 42 58 54 55 72 48 71 30 41 57 ...
## $ week : int 0 0 0 0 0 0 0 0 0 0 ...summary(BPRSL)## X treatment subject weeks
## Min. : 1.00 Min. :1.0 Min. : 1.00 week0 : 40
## 1st Qu.: 90.75 1st Qu.:1.0 1st Qu.: 5.75 week1 : 40
## Median :180.50 Median :1.5 Median :10.50 week2 : 40
## Mean :180.50 Mean :1.5 Mean :10.50 week3 : 40
## 3rd Qu.:270.25 3rd Qu.:2.0 3rd Qu.:15.25 week4 : 40
## Max. :360.00 Max. :2.0 Max. :20.00 week5 : 40
## (Other):120
## bprs week
## Min. :18.00 Min. :0
## 1st Qu.:27.00 1st Qu.:2
## Median :35.00 Median :4
## Mean :37.66 Mean :4
## 3rd Qu.:43.00 3rd Qu.:6
## Max. :95.00 Max. :8
## BPRSL$treatment <- factor(BPRSL$treatment)
BPRSL$subject <- factor(BPRSL$subject)Plotting the BPRSL data
dim(BPRSL)## [1] 360 6ggplot(BPRSL, aes(x = week, y = bprs, group = treatment)) +
geom_line() + aes(linetype = treatment) + scale_x_continuous(name = "week (weeks)", breaks = seq(0, 60, 10)) + scale_y_continuous(name = "treatment (points)") + theme(legend.position = "top")
BPRS_reg <- lm(formula = bprs ~ week + treatment, data = BPRSL)
summary(BPRS_reg)##
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRSL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16In this plot it seems that there are some differencies between the treatment groups, but in the regression model the treatmentgroups dont differ with statistical significance. The flowing time/weeks instead seems to have a statistically significant meaning to the overall decreasing symptoms. The only problem here is that here we concider the variables as independent, witch is FALSE, so this really isnt the truth yet.
Fitting a random intercept model
BPRS_ref <- lmer(bprs ~ week + treatment + (1 | subject), data = BPRSL, REML = FALSE)
summary(BPRS_ref)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9090 24.334
## week -2.2704 0.2084 -10.896
## treatment2 0.5722 1.0761 0.532
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000In this case there isnt any significant differences between the values in the model where we considered the variables independent and this with random intercept.
Random intercept and random slope model
BPRS_ref1 <- lmer(bprs ~ week + treatment + (week | subject), data = BPRSL, REML = FALSE)
summary(BPRS_ref1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7977
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8222 8.0512
## week 0.9609 0.9803 -0.51
## Residual 97.4304 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 2.1052 22.066
## week -2.2704 0.2977 -7.626
## treatment2 0.5722 1.0405 0.550
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000anova(BPRS_ref1, BPRS_ref)## Data: BPRSL
## Models:
## BPRS_ref: bprs ~ week + treatment + (1 | subject)
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1These values are also very similar to the random intercept model, but the chi-square gives this model better fit with statistical significance.
Random intercept and random slope model with interaction
BPRS_ref2 <- lmer(bprs ~ week * treatment + ( week| subject), data = BPRSL, REML = FALSE)
summary(BPRS_ref2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week * treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0768 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.9964 8.0620
## week 0.9687 0.9842 -0.51
## Residual 96.4707 9.8220
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.8856 2.2521 21.262
## week -2.6283 0.3589 -7.323
## treatment2 -2.2911 1.9090 -1.200
## week:treatment2 0.7158 0.4010 1.785
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840anova(BPRS_ref2, BPRS_ref1)## Data: BPRSL
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week * treatment + (week | subject)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4
## BPRS_ref2 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Fitted <- fitted(BPRS_ref2)
BPRSL <- BPRSL %>%
mutate(Fitted)